Final answer:
To find the maximum value of an exponential function, you can follow these steps: rewrite the function, find the derivative, set it equal to zero, solve for x, and plug it back in to find the maximum value. An example is provided.
Step-by-step explanation:
To find the maximum value of an exponential function, you can follow these steps:
- First, rewrite the exponential function in the form f(x) = ae^mx, where a is a constant and m is the exponent.
- Next, take the derivative of the function with respect to x to find the critical points.
- Set the derivative equal to zero and solve for x to find the critical point(s).
- Plug the critical point(s) back into the original function to find the corresponding y-coordinate(s).
- The maximum value of the exponential function will be the highest y-coordinate among the critical point(s) you found.
For example, let's say we have the function f(x) = 2e^(-0.5x). We can find the maximum value by taking the derivative, setting it equal to zero, solving for x, and plugging it back into the original function.
Derivative: f'(x) = -0.5e^(-0.5x)
Critical point: -0.5e^(-0.5x) = 0
x = 0
Plugging x = 0 back into the original function: f(0) = 2e^0 = 2
So the maximum value of the function is 2.